Key to Practice Midterm III

Problem 1

1. False, It could be different on a nonlinear path for example, 5 + xy²/(x² + y²)

2. True:  z = f(u), f_x = f_uu_x = f_u, f_y = f_uu_y = -f_u 

Problem 2
Volume of 2/sqrt3 with dimensions of sqrt2 by 1 by sqrt(2/3)

Problem 3
B.  Begin very high on the positive x-axis.  Fall down the cliff to land at a valley at (1/sqrt2 , 1/sqrt2).  Then climb up an infinite cliff.  Past the y-axis begin at an infinitely low point climbing to a ridge at (-1/sqrt2, 1/sqrt2).  Then slide down to an infinitely low point at the negative y-axis.  On the other side of the y-axis follow the same elevation change as the prior part of the trip.

Problem 4

A.  DNE
B.  0

Problem 5A.  

1. 2x + y
   

2. f_xxx_u + f_xyy_u + f_xx_uu + f_xyx_u + f_yyy_u + f_yy_uu
   

3. 29

Problem 6
0.59 ohms

Problem 7
x = -1 + 2t,     y = 1 + 12t,     z = 2 + 4t